Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #9693 |
478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,56,57)(67,72)(71,79)(73,78)(80,81);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(59,63)(60,61)(64,65)(66,68,70,77,69,74,76,75)(67,71,78,80,72,79,73,81);(55,57)(58,59,61,63,62,65,60,64)(66,67,70,73,69,72,76,78)(68,71,75,80,74,79,77,81)Polycyclic presentation:
2^8,53|a^2=f,b^2=df,c^2=eg,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh,i^b=i^521514 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 68605);